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Unified representation of the family of L-functions
Journal of Inequalities and Applications volume 2013, Article number: 64 (2013)
The aim of this paper is to unify the family of L-functions. By using the generating functions of the Bernoulli, Euler and Genocchi polynomials, we construct unification of the L-functions. We also derive new identities related to these functions. We also investigate fundamental properties of these functions.
AMS Subject Classification:11B68, 11S40, 11S80, 26C05, 30B40.
The theory of the family of L-functions has become a very important part in the analytic number theory. In this paper, using a new type generating function of the family of special numbers and polynomials, we construct unification of the L-functions.
Throughout this presentation, we use the following standard notions , , , . Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real number and ℂ denotes the set of complex numbers. We assume that denotes the principal branch of the multi-valued function with the imaginary part constrained by .
Recently, the first author  introduced and investigated the following generating functions which give a unification of the Bernoulli polynomials, Euler polynomials and Genocchi polynomials:
where ( when ; when ; ; (); ).
For the special values of a, b, k, b and β, the polynomials provide us with a generalization and unification of the classical Bernoulli polynomials, Euler polynomials and Genocchi polynomials and also of the Apostol-type (Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi) polynomials.
Remark 1.1 If we set in (1), we get a special case of the generalized Bernoulli polynomials , that is, the so-called Apostol-Bernoulli polynomials generated by
Remark 1.2 By substituting in (1), we are led to Apostol-Euler polynomials which are defined by means of the following generating function:
Remark 1.3 Setting into (1), we get the Apostol-Genocchi polynomials which are defined by means of the following generating function:
In terms of a Dirichlet character χ of conductor , Ozden et al.  extended and investigated the generating functions of the generalized Bernoulli, Euler and Genocchi numbers and the generalized Bernoulli, Euler and Genocchi polynomials with parameters a, b, β and k. Such χ-extended polynomials and χ-extended numbers are useful in many areas of mathematics and mathematical physics.
Definition 1.4 (Ozden et al. [, p.2783])
Let χ be a Dirichlet character of conductor . Then the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi numbers and the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi polynomials are given by the following generating functions:
where ( when ; when ; ; (); ) and
( when ; when ; ; (); ).
Remark 1.5 Substituting into (2), we are led immediately to the generating function of the generalized Bernoulli numbers which are defined by means of the following generating function:
2 Unification of the L-functions
Our aim in this section is to apply the Mellin transformation to the generating function (3) of the polynomials in order to construct a unification of the various members of the family of the L-functions and to thereby interpolate for negative integer values of n.
Throughout this section, we assume that with and .
By substituting (1) into (2), we obtain the following functional equation:
By using this functional equation, we arrive at the following theorem.
Theorem 2.1 Let χ be a Dirichlet character of conductor f. Then we have
By using (5), we modify (3) as follows:
By using (7), we derive the following result.
Corollary 2.2 Let χ be a Dirichlet character of conductor . Then we have
where the additional constraint is required for the convergence of the infinite integral, which is given in (9), at its upper terminal. By making use of the above integral representation, Ozden et al. [, p.2784 Equation (4.1)] defined the unified zeta function as follows:
By applying the Mellin transformation to the generating function (7), we have the following integral representation of the unified two-variable L-functions :
in terms of the generating function defined in (7). By substituting (9) into ( 11), we obtain
where ( (); ()).
Consequently, by making use of (10) and (12), we are ready to define a two-variable unification of the Dirichlet-type L-functions as follows.
Definition 2.3 Let χ be a Dirichlet character of conductor . For (), we define a two-variable unified L-function by
Remark 2.4 If we substitute into (13), we get the unified L-function
where (, ()).
Remark 2.5 Upon substituting and into (13), we arrive at the interpolation function for twisted generalized Eulerian numbers and polynomials, which is given as follows:
where, for a positive integer r, ξ is the r th root of 1.
Remark 2.6 Substituting into (13), we get a unification of the L-functions
Substituting into (13), we get a unification of the Hurwitz-type zeta function which is given in (10). We also note that both the Hurwitz (or generalized) zeta function
where the Hurwitz-Lerch zeta function is defined by
which converges for (, when ; when ), where as usual
A relationship between the functions and is provided by the next theorem.
Theorem 2.7 Let . Let χ be a Dirichlet character of conductor . Then we have
Proof Substituting , , into (13), we obtain
After some algebraic manipulations, we arrive at the desired result. □
Remark 2.8 Substituting into (13), we have
which interpolates the Apostol-Bernoulli polynomials attached to the Dirichlet character, which are given by means of the following generating functions:
Let f be an odd integer. If we set and into (13), then we have
which interpolate the Apostol-Euler polynomials attached to the Dirichlet character, which are defined by the following generating functions:
By using (15) and (14), we arrive at the following result.
Corollary 2.9 Let . Let χ be a Dirichlet character of conductor . Then we have
Theorem 2.10 Let χ be a Dirichlet character of conductor f. Let n be a positive integer. Then we have
Proof By substituting into (15), we get
By using Theorem 7 in , we get
By substituting (8) into the above, we arrive at the desired result. □
Remark 2.11 The two-variable Dirichlet L-function and the Dirichlet L-function are obvious special cases of the unified Dirichlet-type L-functions defined by (13). We thus have (cf. )
where . By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. We have
where and , the usual generalized Bernoulli number, is defined by (4). The Dirichlet L-function is used to prove the theorem on primes in arithmetic progressions. Dirichlet shows that is non-zero at . Furthermore, if χ is a principal character, then the corresponding Dirichlet L-function has a simple pole at (cf. [6, 7, 9, 18, 24, 27, 28, 30, 31]).
we construct a meromorphic function involving a unified family of L-functions. Therefore, using (16) and (17),
From the above equation, we arrive at the following theorem.
Theorem 3.1 Let . Let χ be a Dirichlet character of conductor f. Then we have
The function is an analytic function at . We now compute the value of this function at this point as follows:
The function is a meromorphic function. This function has simple poles which are
The residues of this function at the simple poles at and are given, respectively, as follows:
where (); (); . Observe that if , then is reduced to the work of Kim .
Relationship between the function and is given as the following result.
Corollary 3.3 Let χ be a Dirichlet character of conductor f. Then we have
We conclude our present investigation by remarking that the existing literature contains several interesting generalizations and extensions of the Hurwitz-Lerch zeta function , Hurwitz zeta function and L-function (cf. [1–30]); see also the references cited in each of these earlier works.
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Dedicated to Professor Hari M Srivastava.
Both authors are partially supported by Research Project Offices Akdeniz Universities and the Commission of Scientific Research Projects of Uludag University Project number UAP(F) 2011/38 and 2012/16. We would like to thank referees for their valuable comments.
The authors declare that they have no competing interests.
Both authors completed the paper together. Both authors read and approved the final manuscript.
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Ozden, H., Simsek, Y. Unified representation of the family of L-functions. J Inequal Appl 2013, 64 (2013). https://doi.org/10.1186/1029-242X-2013-64
- Bernoulli numbers
- Bernoulli polynomials
- Euler numbers
- Euler polynomials
- Genocchi numbers
- Genocchi polynomials
- Dirichlet L-functions
- Hurwitz zeta function
- Riemann zeta function